Mathematics 700 Homework.

1. This problem is a warm up for some of the machinery that we will need to understand normal forms for similarity which is the high point of the class. Let V be a vector space and let U, W be subspaces of V so that U +W = V and U ∩W = {0}. Then you have shown in an old homework (page 40, problem 9) that every v ∈ V can then be uniquely expressed as v = u + w where u ∈ U and w ∈ W. In this case we write V = U ⊕ W and say that V is the direct sum of U and W. (To be redundant (but explicit) if from now on you hear that V is a direct sum of U and W (or see the formula V = U ⊕ W) then you are immediately to think that U and W are subspaces of V with U + W = V, U ∩ W = {0} and also remember that in particular this means that every v ∈ V can be uniquely written as v = u + w with u ∈ U and w ∈ W.)

(a) If V = U ⊕ W and u_{1}, …, u_{k} is a basis of U and w_{1}, …, w_{m} is a basis of W then show u_{1}, …, u_{k}, w_{1}, …, w_{m} is a basis of V. Thus

dim V = dim U + dim W.

Remark: This also follows from the formula dim(W_{1} + W_{2}) + dim(W_{1} ∩ W_{2}) = dim W_{1} + dim W_{2}.

(b) If V = U ⊕ W then define a linear map P : V → V by

P v = u where v = u + w with u ∈ U and w ∈ W

P is called the projection of V onto U with kernel W.

i. Show Image P = U and Ker P = W.

ii. Show that P^{2} = P.

iii. Show P u = u if and only if u ∈ U.

(c) Let P : V → V be a linear map that satisfies P^{2} = P. (Remark: Linear maps P satisfying P^{2} = P are called projections. They occur very naturally in both algebra and analysis.)

i. Show

V = Image P ⊕ Ker P.

(Hint: Every v ∈ V can be written as v = P v + (v − P v).) ii. Let k = rank P = dim Image P and let v_{1}, …, v_{k} be a basis for Image P and v_{k+1}, …, v_{n} (where n = dim V) be a basis of Ker P. Then by the first problem above v_{1}, …, v_{n} is a basis of V. What is the matrix of P in this basis?

(d) Let T : V → V be a linear map that satisfies the equation T^{2} = I (where I is the identity transformation.) Then as a variant on the last problem

i. Show

V = Ker(T − I) ⊕ Ker(T + I).

ii. If v_{1}, …, v_{n} is a basis if V so that v_{1}, …, v_{k} is a basis of Ker(T − I) and v_{k+1}, …, v_{n}

is a basis of Ker(T + I) then what is the matrix of T in this basis? iii. Can you generalize to T that satisfy other polynomial equations? For example what if T satisfies T^{3} = T? Remark: If you don’t see a generalization don’t spend much time on this. If you do see what is going on it is worth pursuing as it will give you a leg up on much of what we will be doing.

Quotient Spaces

Let V be a vector space over the field F and W a subspace of V. Then define an equivalence relation ∼_{W} by

v_{1} ∼_{W} v_{2} if and only if v_{2} − v_{1} ∈ W.

Problem 1 Show that this is an equivalence relation.

Denote by [v]_{W} the equivalence class of v ∈ V under the equivalence relation ∼_{W}. That is

[v]_{W} := {u ∈ V : u ∼_{W} v}.

Problem 2 Show [v]_{W} = v + W where v + W = {v + w : w ∈ W }. Let V/W be the set of all equivalence classes of ∼_{W}. That is

V/W := {[v]_{W} : v ∈ V } = {v + W : v ∈ V }.

The equivalence class [v]_{W} = v + W is often called the coset of v in V/W. Problem 3 Let V = R^{2} and let W be the subspace of points of V of points (x, y) with y = −2x. Then draw pictures of the coset of (1, 1) in V/W and the coset of (3, −2) in V/W. What is a geometric description of the coset of v ∈ R in V/W? Define a sum and scalar multiplication in V/W by

[v_{1}]_{W} + [v_{2}]_{W} := [v_{1} + v_{2}]_{W} c[v]_{W} := [cv]_{W}

where v_{1}, v_{2}, v ∈ V and c ∈ F. Problem 4 Show this is well defined. The term well defined is used in mathematics to mean “is independent of the choices made in the definition”. In this particular case this means you need to show

[v_{1}]_{W} = [v′_{1}]_{W} and [v_{2}]_{W} = [v′_{2}]_{W}  implies  [v_{1} + v_{2}]_{W} = [v′_{1} + v′_{2}]_{W}

and

[v]_{W} = [v′]_{W}  implies  [cv]_{W} = [cv′]_{W} .

Proposition 0.1 With these operations V/W is a vector space.

Problem 5 Prove this.

Problem 6 If V is finite dimensional then what is the dimension of V/W in terms of dim V and dim W? Prove your answer is correct.

Problem 7 In the example of problem 3 draw some pictures of cosets v_{2} + W and v_{2} + W what their sum (v_{1} + W) + (v_{2} + W) and the linear combination 2(v_{1} + W) − 3(v_{2} + W) for a few choices of v_{1} and v_{2}.

Other Problems

Note that what I have been calling eigenvalues is what the book calls characteristic values.

1. Let

A := [ \begin{matrix}a & b \\ c & d \end{matrix} ]

be a two by two matrix. As usual trace A = a + d, det A = ad − bc. Set f(x) = x^{2} − (trace A)x + det A. Show f(A) = 0. Show the minimal polynomial of A divides f(x).

2. Let A be the matrix

A := [ \begin{matrix}0 & 1 \\ −2 & 3 \end{matrix} ]

(a) Use the last problem to find the minimal polynomial of A.

(b) Find the eigenvalues of A.

(c) Find a formula for A^{n}. Hint: Find a basis of R^{2} of eigenvectors of A. In this basis it is easy to compute powers of A. Do the computation in this basis and then change back to the standard basis.

3. Define a sequence x_{0} = 0, x_{1} = 1 and x_{k+1} = 3x_{k} − 2x_{k−1} for k ≥ 1. Let A be the matrix in the last problem. Then show

A [ \begin{matrix}x_{k−1} \\ x_{k} \end{matrix} ] = [ \begin{matrix}x_{k} \\ x_{k+1} \end{matrix} ]  and  A^{k} [ \begin{matrix}0 \\ 1 \end{matrix} ] = [ \begin{matrix}x_{k} \\ x_{k+1} \end{matrix} ] .

Now find a formula for x_{k}. Hint: By the last problem you have a forumla for A^{k}.